In this paper we will focus on the case \(\Sigma /\varphi =\mathbb {P}^1\) , so that \(M_{\varphi }\)  has \(b_1=1\)  and the singularities on the circle of reducible solutions \(\mathbb {T}\)  are as mild as one could hope; we will see how to achieve transversality while still maintaining a concrete understanding of the underlying geometry by adding suitable perturbations. In order to do so, we will use the same pertubations as those employed in [1]} to study the genus one case; on the other hand, our arguments are significantly more delicate (and indeed generalize those appearing there) as we cannot rely on the existence of a flat metric on the mapping torus.
